Properties of GKM graphs

is_compact(G::AbstractGKM_graph) -> Bool

Return true if G is compact, i.e. all flags at all vertices are associated with edges (no standalone flags).

Example

julia> G = projective_space(GKM_graph, 2)
GKM graph with 3 nodes, valency 2 and axial function:
2 -> 1 => (-1, 1, 0)
3 -> 1 => (-1, 0, 1)
3 -> 2 => (0, -1, 1)

julia> is_compact(G)
true

julia> M = G.M;

julia> add_standalone_flag!(G, 1, gens(M)[1]);

julia> add_standalone_flag!(G, 2, gens(M)[2]);

julia> add_standalone_flag!(G, 3, gens(M)[3]);

julia> is_compact(G)
false

betti_numbers(G::AbstractGKM_graph) -> Vector{Int64}

Return the array betti_numbers such that betti_numbers[i+1] is the 2i-th combinatorial Betti number for i from 0 to the valency of G, as defined in [GZ01, section 1.3].

Examples

julia> H6 = gkm_graph_of_toric(hirzebruch_surface(NormalToricVariety, 6));

julia> betti_numbers(H6)
3-element Vector{Int64}:
 1
 2
 1

valency(G::AbstractGKM_graph) -> Int64

Return the valency of G, i.e. the number of flags at each vertex.

Example:

The valency of the GKM graph of $\mathbb{P}^3$ is 3, since all of the fixed points $[1:0:0:0], \dots, [0:0:0:1]$ are connected to each other via some $T$-invariant $\mathbb{P}^1$'s. For example, $[1:0:0:0]$ and $[0:1:0:0]$ are connected by $\{[x:y:0:0] : x,y\in\mathbb{C}\}$.

julia> valency(projective_space(GKM_graph, 3))
3
julia> valency(grassmannian(GKM_graph, 2, 4)) # The Grassmannian of 2-planes in C^4
4
julia> valency(flag_variety(GKM_graph, [1, 1, 1, 1])) # The variety of full flags in C^4
6

rank_torus(G::AbstractGKM_graph) -> Int64

Return the rank of the torus acting on G. That is, the rank of the character group.

Examples

By default, the torus acting on $\mathbb{P}^n$ is $(\mathbb{C}^\times)^{n+1}$, acting by rescaling the homogeneous coordinates.

julia> P3 = projective_space(GKM_graph, 3);

julia> rank_torus(P3)
4

Taking products adds the rank:

julia> H6 = gkm_graph_of_toric(hirzebruch_surface(NormalToricVariety, 6));

julia> rank_torus(H6)
4
julia> rank_torus(H6 * P3)
8

is2_indep(G::AbstractGKM_graph) -> Bool

Return true if G is 2-independent, i.e. the weights of every two flags at a vertex are linearly independent.

is3_indep(G::AbstractGKM_graph) -> Bool

Return true if G is 3-independent, i.e. the weights of every three flags at a vertex are linearly independent.

Example

The weights of $\mathbb{P}^3$ at the fixed point $[1:0:0:0]$ are $\{t_i-t_0:i\in\{1, 2, 3\}\}$, which are linearly independent over $\mathbb{C}$.

julia> is3_indep(projective_space(GKM_graph, 3))
true

The variety of complete flags in $\mathbb{C}^3$ is an example of a GKM graph that is not 3-independent:

julia> G = flag_variety(GKM_graph, [1, 1, 1])
GKM graph with 6 nodes, valency 3 and axial function:
13 -> 12 => (0, -1, 1)
21 -> 12 => (-1, 1, 0)
23 -> 13 => (-1, 1, 0)
23 -> 21 => (-1, 0, 1)
31 -> 13 => (-1, 0, 1)
31 -> 21 => (0, -1, 1)
32 -> 12 => (-1, 0, 1)
32 -> 23 => (0, -1, 1)
32 -> 31 => (-1, 1, 0)

julia> is3_indep(G)
false

is_strictly_nef(G::AbstractGKM_graph) -> Bool

Return true if and only if the Chern numbers of all curve classes corresponding to edges of the GKM graph are strictly positive.

Examples

julia> F3 = flag_variety(GKM_graph, [1,1,1]);

julia> print_curve_classes(F3)
13 -> 12: (0, 1), Chern number: 2
21 -> 12: (1, 0), Chern number: 2
23 -> 13: (1, 1), Chern number: 4
23 -> 21: (0, 1), Chern number: 2
31 -> 13: (1, 0), Chern number: 2
31 -> 21: (1, 1), Chern number: 4
32 -> 12: (1, 1), Chern number: 4
32 -> 23: (1, 0), Chern number: 2
32 -> 31: (0, 1), Chern number: 2

julia> is_strictly_nef(F3)
true

julia> H5 = gkm_graph_of_toric(hirzebruch_surface(NormalToricVariety, 5))
GKM graph with 4 nodes, valency 2 and axial function:
2 -> 1 => (1, 0, -1, 0)
3 -> 2 => (5, 1, 0, -1)
4 -> 1 => (0, 1, 5, -1)
4 -> 3 => (-1, 0, 1, 0)

julia> print_curve_classes(H5)
2 -> 1: (-5, 1), Chern number: -3
3 -> 2: (1, 0), Chern number: 2
4 -> 1: (1, 0), Chern number: 2
4 -> 3: (0, 1), Chern number: 7

julia> is_strictly_nef(H5)
false

fano_index(G::AbstractGKM_graph) -> ZZRingElem

Return the Fano index of the GKM graph, which is the gcd of the first Chern numbers of all its edges.

Examples

julia> P2 = projective_space(GKM_graph, 2);

julia> fano_index(P2)
3

julia> F3 = flag_variety(GKM_graph, [1, 1, 1]);

julia> fano_index(F3)
2

index_periodic_betti(G::AbstractGKM_graph) -> Vector{Int64}

Return the index-periodic Betti numbers of a compact GKM graph.

For a GKM graph with Fano index $p$, this function computes the sums of Betti numbers grouped by their residue class modulo $p$. Specifically, the $i$-th entry (for $i$ from $1$ to $p$) contains the sum of all Betti numbers $b_j$ where $j \equiv i$ (mod $p$).

This is as defined in [BGL+25, before Theorem 2.2].

Examples

julia> P2 = projective_space(GKM_graph, 2);

julia> index_periodic_betti(P2)
3-element Vector{Int64}:
 1
 1
 1

QH_ss_check_GLLXBR(G::AbstractGKM_graph)::Bool

Return true if $G$ satisfies conditions (1) and (2) in [BGL+25, Theorem 2.2].